The Federal Aviation Administration has approved Boeing’s plan to test its redesigned 787 Dreamliner battery system, a critical step toward certifying the fix and getting the grounded jets back in the air.

Now Boeing, which builds the 787 in North Charleston and Everett, Wash., has to prove its solution to the vexing battery problem works.

“There’s no question that this is good news for Boeing today, but there’s no way to tell what yard line they’re on,” said Scott Hamilton, who has been closely following the latest 787 saga from his Issaquah, Wash.-based aviation consulting firm Leeham Co. “One would hope they’re marching downfield. There’s just too much that we don’t know.”

The approval, which the government regulator announced Tuesday after weeks of review, allows Boeing to perform flight tests on two Everett-made planes with the lithium ion battery fix in place.

“This comprehensive series of tests will show us whether the proposed battery improvements will work as designed,” U.S. Transportation Secretary Ray LaHood said in a statement. “We won’t allow the plane to return to service unless we’re satisfied that the new design ensures the safety of the aircraft and its passengers.”

Boeing CEO Jim McNerney said in a prepared statement that safety is Boeing’s top priority, and that teams have been working “around the clock” to solve the battery mystery.

“Today’s approval from the FAA is a critical and welcome milestone toward getting the fleet flying again and continuing to deliver on the promise of the 787,” he said.

The FAA grounded the technologically advanced but historically troubled 787 on Jan. 16 after a pair of battery overheating incidents.

The root cause of the malfunctions has not been determined, but Boeing has developed what its commercial airplanes chief called a comprehensive and permanent solution to the issue, and presented that to the FAA on Feb. 22.

It involves “a redesign of the internal battery components to minimize initiation of a short circuit within the battery, better insulation of the cells and the addition of a new containment and venting system” to prevent fires, according to Tuesday’s FAA statement.

In its statement, Boeing noted that the upgrade’s “enhanced production and testing processes include more stringent screening of battery cells prior to battery assembly,” and tightening the system’s voltage range.

The certification plan proposed by Boeing “establishes specific pass/fail criteria, defines the parameters that should be measured, prescribes the test methodology and specifies the test setup and design,” according to the FAA.

The flight tests will seek to validate the aircraft instrumentation for the battery and battery enclosure testing, in addition to product improvements for other systems.

Boeing said the two planes are the fifth test-flight plane and line number 86. According to the website All Things 787, LN 86 is destined for LOT Polish Airlines, which is one of the hardest-hit and most vocal 787 customers.

The FAA noted that its engineers will be “closely involved in all aspects of the process.”

The FAA statement also noted its airworthiness directive from January, which required operators to temporarily cease 787 operations, is still in effect, and that its comprehensive review of the 787 program, announced days before the grounding in January, will continue.

Boeing had delivered 50 787 Dreamliners, including four made in North Charleston, when the battery incidents, first on the ground in Boston, then in the air over Japan, led to a global grounding.

Certification and return to flight is hardly a foregone conclusion after Tuesday’s FAA approval, according to Hamilton.

“The biggest hurdle will be to pass all the pass/fail tests,” he said. “This is still a challenging effort for Boeing to validate all the lab work that they’ve done. So there’s still some mountains to climb yet.”

He said it’s “anybody’s guess” how long it will be now before the FAA allows the fleet to return to flight, because the exact series of tests hasn’t been publicized.

“We don’t know how long it will take to do the tests in the lab, we don’t know how long it will take to do the tests in the air,” he said.

Meanwhile, the National Transportation Safety Board is still trying to get to the bottom of the Jan. 7 incident in Boston, and Japanese authorities are investigating the Jan. 16 incident there.

For its part, Boeing is still studying the issue while producing five 787s per month, with plans to double that rate to 10 per month by the end of this year. Four 787s sit completed on the flight line in North Charleston, with many more poised for delivery in Everett.

Reach Brendan Kearney at 937-5906 and follow him on Twitter at @kearney_ brendan.